Control of deposition and other processes

ABSTRACT

Material is incrementally deposited using material directed toward a deposition zone. The scan path of the directed material is controlled according to a path plan derived to reduce derivation from an ideal uniform temperature profile for the deposition during the deposition process. A path plan having angled scan passes that intersect (or overcross one another), for example in a mirrorbox path plan, is preferred.

CROSS-REFERENCE TO RELATED APPLICATION

This application is a 371 of PCT/GB02/00983, filed Mar. 5, 2002.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates primarily to control for processesinvolving deposited material (such as for example molten metal sprayingprocesses).

2. State of the Art

WO-A-96/09421 discloses a technique for spraying molten metal(particularly steel) to produce self supporting articles. In the processdisclosed it is clear that for a practically realisable process,accurate control of the temperature of the sprayed metal droplets and/orthe temperature of the already deposited material is important. Suchconsiderations are also relevant to spraying of other materials andother deposition processes. Additionally other parameters for spraydeposition processes require monitoring regulation and control.

The spray forming process deposits molten metal (typically from electricarc spray guns) onto a substrate (typically a ceramic substrate) to forma metal shell that accurately reproduces the topography of the ceramicsubstrate.

The molten metal is typically produced in the guns by direct currentarcing between two oppositely charged wires made of the metal beingsprayed. The arcing causes the wire tips to melt and a high-pressureinert gas stream continuously strips molten material from the arc,atomising it into a spray of droplets. The gas stream carries thedroplets to the surface of the object where they are deposited. Wire iscontinuously fed to the arc gun to maintain the flow of sprayed metaland the amount of metal that is deposited can be adjusted by changingthe feed rate of the wire. The droplet spray from the guns is scannedover the surface of the ceramic substrate by a robot in apre-determined, repetitive manner, referred to as the “path plan”.

The guns act not only as source of material but also as source of heatbecause the molten droplets transfer their heat to the spray formedmetal shell as they cool and solidify to build up a solid metal shell.An important feature of the process described in WO-A-96/09421 is thatit relies on the metal droplets undergoing prescribed phasetransformations as they cool after being deposited on the surface of thesprayform. These phase transformations offset the natural contraction ofthe metal as it cools, allowing the dimensional accuracy of thesprayform to be maintained. In order to ensure that the requiredtransformations occur, accurate regulation of the thermal history of thesprayed material is necessary. One method of regulating the thermalhistory is to ensure that the temperature of the surface at the pointwhere the spray was deposited passes through a given temperature at aspecific time after deposition. A system for regulating the thermalhistory of the deposited material has been proposed that adjusts one ormore parameters including the height, velocity and path of the robot andthe orientation of the guns. These adjustments are made relative tonominal or reference values for these variables and the purpose of thecurrent invention is to specify a nominal path for the robot over thesprayform that will minimise the variations in temperature over thesurface.

SUMMARY OF THE INVENTION

According to a first aspect, the present invention provides a system forincrementally depositing material, which system comprises:

-   -   delivery means for directing material toward a delivery zone;    -   control means for controlling operation of the delivery means,        the control means controlling the deposition according to a        derived path plan predicted to minimise/reduce deviation from an        ideal uniform temperature profile during the deposition process.

According to a second aspect, the invention provides a control systemfor deposition apparatus, the control system controlling the depositionaccording to a derived path plan predicted to minimise/reduce deviationfrom an ideal uniform temperature profile during the deposition process.

According to a further aspect the invention provides a method ofproducing an article by a deposition process, the method comprisingdirecting material toward a delivery zone and controlling the depositionaccording to a derived path plan predicted to minimise/reduce deviationfrom an ideal uniform temperature profile during the deposition process.

The material is typically delivered in flight, preferably asvapour/molten droplets. Typically the material may be delivered by spraydelivery means. Molten droplets of the material are typically atomisedin a conveying gas.

The delivery means is typically operable to produce a scanning ortraversing pattern of material deposition or flight delivery over thedeposition zone; the control means beneficially operates (at leastinitially) to the predetermined path plan having predetermined scan ortraverse rate or scan movement direction.

The path plan preferably comprises a predetermined path plan derived byconsidering spatial modes and selecting spatial modes to optimise thescan launch angle and/or path plan length preferably without excitinglower order modes. The scan path plan preferably reflects at boundariesto form an overcrossing pattern at the deposition zone.

The path plan may comprise a repeating pattern returning to a startpoint following a plurality of scan passes over the deposition zone.Alternatively the path plan may comprise a non-repeating pattern, anartificial correction step may return the path to a common path pointfollowing a finite number of scan passes.

The predetermined path plan is beneficially derived in a process(preferably a computer software run process) in which one or more of thefollowing input considerations are accredited:

-   -   optimisation criteria selected;    -   maximum acceptable deviation from desired thermal profile;    -   dimensions of the deposition zone    -   size/dimensions of deposition footprint;    -   mass deposition criteria;    -   scan velocity.

Beneficially a scan angle is set in which:

-   -   having regard for the footprint of the spray gun, {tilde over        (f)}(x, y, t), the coefficients, {circumflex over (b)}_(m,n),        are determined (typically when the gun effecting deposition is        centred over the deposition zone surface);    -   Upper bounds, M and N are determined, such that {{circumflex        over (b)}_(m,n)≈0:m>M; n>N};    -   Integers are selected, μ≦M and υ≦N, such that μ≧υ and μ and υ        have no common factors;    -   Scan angle set to

${\tan\;\psi} = \frac{\mu\; L_{y}}{v\; L_{x}}$

-   -   Search over all modes, {m=1,2, . . . ,M,n=1,2, . . . ,N}, to        ensure that all q_(m,n)(t) satisfy the optimisation criterion        for this scan angle.

If the criterion is not satisfied, increase υ and/or μ and repeatpreceding steps (from ‘Integers are selected’ step);

-   -   If the criterion is satisfied, check that path satisfies mass        deposition criterion;    -   If the mass deposition criterion is not satisfied, increase υ        and/or μ and repeat preceding steps (from ‘integers are        selected’ step).    -   If the mass deposition criterion is satisfied, use scan angle, ψ        to generate robot path and download to control scan (download to        scanning robot).

The system according to the invention is particularly suited to theproduction of articles in which localised differences in thermalconditions and/or thermal history can lead to differential thermalcontraction and distortion. The optimised path plan selection andcontrol enables the spraying regime for such large articles to beclosely and accurately regulated.

UK Patent Application 0026868.0 (the entirety of which is incorporatedherein by reference) relates to controlling deposition processesincluding the thermal profile of deposited material using real timemonitoring of parameters (including thermal parameters) to ensure that adesired thermal history has occurred for deposited material.

The present invention is of benefit in its own right as providing forthermal history control by providing an optimised path plan fordeposition. Alternatively, the present invention provides a usefuladjunct for control processes such as that described in UK PatentApplication 0026868.0 because a deposition process can initially be setup to run in accordance with the path plan derived according to thepresent invention and subsequently feedback monitored control input canbe utilised if desired for more accurate or sophisticated control.

The system can be used to spray to a predetermined desired temperatureprofile at which different surface zones may be maintained at differenttemperatures at different times during the spraying process.

The present invention is also of benefit for controlling projecteddelivery/deposition processes (such as spraying) of materials havingother parameters which are time variable (particularly followingdeposition). Examples of such situations and processes are heat flow,fluid flow, diffusion, decomposition and curing. This list isnon-exhaustive. The invention may for example be utilised in processessuch as paint spraying where fluid flow may occur following deposition.

According to a further aspect, the invention therefore provides A systemfor incrementally depositing material, which system comprises:

-   -   delivery means for directing material toward a deposition zone;    -   control means for controlling operation of the delivery means,        the control means controlling the deposition according to a        derived scan path plan predicted to minimise/reduce deviation        from an ideal uniform parameter profile for the deposit during        the deposition process.

According to a further aspect, the invention therefore also provides acontrol system for deposition apparatus, the control system controllingthe deposition according to a derived scan path plan predicted tominimise/reduce deviation from an ideal uniform parameter profile forthe deposit during the deposition process.

The invention as defined is applicable to minimise deviation from theideal value for a parameter of the deposited material that has atendency to vary over time. For example, the thickness of spray paintmaterial may vary over time as the paint flows at the deposition zone.The preferred features of the invention in relation to deposittemperature profile optimisation may also be preferred in relation tooptimisation of parameters for other spray deposited materials orprocesses. Particularly, scan path plans as defined in the claims having‘mirrorbox’ or traversing scan passes as defined will improve theresultant deposition characteristics.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention will now be further described in specific embodiments, byway of example only, and with reference to the accompanying drawings, inwhich:

FIG. 1 is a schematic representation of a system according to theinvention.

FIG. 2 is a representation of a path taken by robot as it scans acrossthe surface of the sprayform deposition zone.

FIG. 3 shows normalised peak amplitude of different spatial modesplotted against scan velocity.

FIG. 4 shows normalised peak amplitude of different modes for a Gaussianfootprint of width L_(x)/20 and scan angle ψ=56.13°.

FIG. 5 shows normalised peak amplitude of different modes for a Gaussianfootprint of width L_(x)/20 and scan angle ψ=72.9°.

FIG. 6 shows normalised peak amplitude of different modes plottedagainst scan angle for an impulsive footprint when the gun velocity is0.2 m s⁻¹. The legend shows the angle at which the peak amplitude occursfor each mode.

FIG. 7 shows normalised peak amplitude of different modes plottedagainst scan angle for a Gaussian footprint of width L_(x)/20 when thegun velocity is 0.2 m s⁻¹. The legend shows the angle at which the peakamplitude occurs for each mode.

FIG. 8 shows normalised peak amplitude of different modes plottedagainst scan angle for a Gaussian footprint of width L_(x)/5 when thegun velocity is 0.2 m s⁻¹. The legend shows the angle at which the peakamplitude occurs for each mode.

FIG. 9 is a schematic representation of an exemplary path that resultsin a good thermal profile, but does not repeat.

FIG. 10 is a schematic representation of an exemplary poor path thatamplifies a particular mode, in this case q_(5,3)(t).

FIG. 11 is a schematic representation of an exemplary closed path thatavoids amplifying low order modes and results in a flat thermal profile.

FIG. 12 is a flowchart for optimisation procedure in accordance with theinvention.

FIG. 13 is a thermal image of a poor “mirrorbox” scan path plan.

FIG. 14 is a thermal image of a good “mirrorbox” scan path plan.

FIG. 15 is a thermal image of a raster scan path plan.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

Referring to FIG. 1, in the embodiment described here the systemconsists of a single spray gun (1) spraying molten steel, mounted on a6-axis industrial robot (2). The robot moves the spray gun over aceramic former (3) and the metal deposited in the spray (4) builds up ametal shell. The temperature profile on the surface is recordedperiodically by a thermal imaging camera (5). A computer (6) determinesthe path to be followed by the gun and downloads it to the robot.

In the invention described in UK Patent Application 0026868.0,adjustments are made to the path of the robot and to parameters such asthe wire feed rate to achieve the desired thermal profile. In theinvention described here, the height, robot velocity, gun orientation,robot path and wire feed rate are kept constant, but the robot velocityand path are chosen to minimise the variations in the thermal profileover the surface. This has two main purposes:

-   -   1. if no further control is applied, then maintaining the        required uniform thermal profile over the surface minimises the        net stresses and/or distortion across the sprayform, provided        that the appropriate spray conditions are used;    -   2. if control of thermal history is applied as described in UK        Patent Application 0026868.0, then adjustments real time to the        robot velocity, robot height, path etc., will be made relative        to the optimal, predetermined, nominal path.

Consider spraying onto a flat rectangular sprayform of dimension L_(x)in the x direction and L_(y) in the y direction (other geometries willbe described below). Viewed from above, FIG. 2 shows the path (pathplan) taken by the spray from the gun (1) as it tracks across thesurface, at constant velocity, ν. The robot starts from a point (A) onone edge of the sprayform and tracks across the sprayform at an angle ψ,to this edge, until it reaches the opposite edge at point B, where theycomponent of the velocity of the robot is reversed, so that the robotturns round and scans back at the same angle, ψ, to the edge. When therobot reaches the end of the sprayform at point C, the x component ofthe velocity of the robot is reversed and robot moves back in thenegative x direction, making an angle of 90°-ψ to the edge.

In order to program the robot movements, it is necessary for the path toconsist of a finite number of moves and the ideal path (from theprogramming point of view) is for the path to end at point A following afinite number of “reflections” at the edges. Under these circumstances,this “closed” path can be repeated either until a new path is determinedor until spraying is complete. If the path does not pass through pointA, then it is necessary to stop the robot at A′, a point close to A,move the robot to point A and then restart the robot along the path.Because the path consists of a series of reflections when the robotreaches the edge of the sprayform, it is referred to as a “mirrorbox”pattern.

If the distance from the gun to the surface and the orientation of thegun remain constant, then for a flat surface, the shape of the“footprint”, or thermal flux, striking the sprayform remains fixed asthe robot moves the gun over the surface. The shape of the footprint inthe current embodiment is a 2-dimensional Gaussian function. For a giventhermal footprint, the invention determines the path (path plan) thatminimises the deviations in the thermal profile over the surface byfinding the optimal scan angle, ψ, and scan velocity, ν. For ease ofexposition, the current embodiment considers the case where L_(x)≧L_(x)and 45 °≦ψ<90°, although the cases where L_(x)<L_(x) and/or ψ<45° can beanalysed by the same approach.

Mass is deposited onto the sprayform by the spray gun as it is movedover the surface by the robot. The invention describes the path (pathplan) that the gun should follow in order to minimise the thermalvariations over the surface of the sprayform.

According to one aspect, the invention requires knowledge of the thermal“footprint” of the gun, which describes the rate of heat deposited perunit area by the spray gun over the surface of the sprayform. Althoughthe shape of the footprint remains constant, its location changes withtime as the gun is moved over the sprayform. The present techniqueexpresses the thermal footprint in terms of a 2-dimensional Fourierseries, which describes the footprint as a weighted sum of 2-dimensionalsinusoidal spatial components. The coefficients of this weighted sum aredenoted by b_(m,n)(t), where m and n are used to index the frequency ofthe spatial harmonics in the x and y dimensions, respectively. Thecoefficients, b_(m,n)(t), vary with time as the spray gun is moved overthe surface.

Because the thermal footprint is a smooth function the surface ofsprayform (i.e it does not contain abrupt changes), the magnitude of thecoefficients, b_(m,n)(t), tends to zero as m and/or n become large,irrespective of the location of the spray gun. This shows that thethermal effect of the gun is concentrated in the low order spatialmodes, i.e. those modes associated with low spatial harmonics. For aclosed spray path, where the path that repeats itself after a fixed timeinterval, the launch angle (i.e the angle that the spray path makes withone edge of the sprayform), ψ, satisfies

${\tan\;\psi} = {\frac{m^{\prime}}{n^{\prime}}\frac{\; L_{y}}{\; L_{x}}}$where m′ and n′ are two integers. The optimal launch angle is determinedby choosing the smallest pair of integers, m′ and n′, such thatb_(m′,n′)(t) is negligible throughout the spray path. Choosing thisvalue of y avoids exciting those thermal modes for which b_(m′,n′)(t)are non-zero. Although any value of m′ and n′ for which b_(m′,n′)(t) isnegligible could be used, choosing the smallest possible values shortensthe length of the path, which simplifies the programming of the robotpath. It is often necessary to minimise variations in the massdeposition, as well as variations in temperature, but because thethermal footprint is highly correlated with the mass depositionfootprint, it is likely that an optimising the launch angle for eventemperature deposition also optimises the deposition of mass.

The variations in temperature can be quantified in terms of its standarddeviation at points over the entire surface. The effect of basing thepath plan on the optimal launch angle y is to ensure that this standarddeviation remains low throughout the path plan. It is possible to find alocation for the spray gun during a non-optimal path plan, where thestandard deviation of the temperature profile is less than the standarddeviation achieved by the optimal path plan. However, for thenon-optimal path, a low standard deviation at one location is offset bymuch larger standard deviations at other points along the path. Thebenefit of the optimal path is that the temperature profile has a lowstandard deviation throughout the path.

EXPERIMENTAL RESULTS

Three experiments were performed under the same spraying conditions butwith different path plans. The spray guns were set at a distance of 160mm from the surface of the sprayform and the robot moved at a constantvelocity of 200 mm.s⁻¹. The guns each deposited mass at 1.8 g.s⁻¹ onto asquare ceramic of dimensions 12 inches by 12 inches. The guns followed afixed path plan which covered an area of 15 inches by 15 inches. Thevariations in the thermal profile were recorded by taking an image usinga thermal imaging camera, one quarter of the way through each repeat ofthe path plan. From the recorded thermal images, the standard deviationof the temperature at each pixel was calculated.

Three different path plans were compared:

-   -   1. “Bad” mirrorbox (FIG. 13)—a path plan with a “poor” launch        angle of 77.47° for the 15 inches by 15 inches pattern, which        excites low order spatial modes associated with the thermal        footprint of the guns    -   2. “Good” mirrorbox (FIG. 14)—a path plan with an optimised        launch angle of 75.07 for the 15 inches by 15 inches pattern        size that avoids the exciting the low order spatial modes        associated with the thermal footprint of the guns    -   3. Raster pattern (FIG. 15) of size 15 inches by 15 inches where        the guns scan across the sprayform in a direction parallel to        one edge. When the guns reach the edge of the spray pattern,        they are moved a short distance parallel to the other edge and        then scan back across the sprayform parallel to the original        track, but in the opposite direction. This is repeated until the        guns reach the edge of the spray pattern when the path is        reversed. This spray path is commonly used in spraying        operations.

The image taken during the 14^(th) complete scan for each path plan waschosen as a typical result and analysed. The images for the badmirrorbox pattern is shown in FIG. 13, while FIG. 14 shows thecorresponding images for the good mirrorbox pattern. The correspondingimage for the raster path plan is shown in FIG. 15. In each case, thelighter areas are the areas where the temperature of the surface ishigher then the average temperature, while the darker areas correspondto regions where the sprayform is cooler than the average temperature.The images were analysed to determine the mean temperature over thesprayform, together with the standard deviation about the mean of thetemperatures associated with the pixels in the mean.

Mean Standard Deviation Path Temperature of Temperature Bad mirrorbox256.4° C. 16.5° C. Good mirrorbox 254.9° C. 12.2° C. Raster pattern260.3° C. 28.5° C.

The results show that the good mirrorbox has the thermal profile withthe lowest standard deviation, indicating that it is the best path touse to minimise the variations in temperature over a scan. The benefitsof the optimal path can also be seen, by examining a sequence of thermalimages. The images in the raster sequence alternate between a lowstandard deviation and very high standard deviation, depending on thepoint in the scan where image is taken. By contrast, the good mirrorboxpattern maintains a low standard deviation throughout the scan and thereis little difference in standard deviation of the images whenever theyare taken.

Procedure for Determining Optimal Path

The procedure (shown in FIG. 12) for determining the optimal path is:

-   -   1. Choose optimisation criterion and maximum acceptable level of        deviation from the desired thermal profile.

2. Input the dimensions of the surface, L_(x), L_(x) and the scanvelocity, ν.

3. Using the thermal footprint of the spray gun, determine thecoefficients, b_(m,n), when the gun is in the center of the sprayform.

4. Determine upper bounds, M and N, such that b_(m,n) 0 for m>M and n>N

-   -   5. Choose integers, m and n, such that m³ n and m and n have no        common factors.

6. Set scan angle, y, to

${\tan\;\psi} = {\frac{\mu}{v}\frac{\; L_{y}}{\; L_{x}}}$

-   -   7. Search over all modes, {(n=1, 2, . . . , M, n=1, 2, . . . ,        N}, to ensure that the optimisation criterion is satisfied for        this scan angle.    -   8. If the criterion is not satisfied, increase n and/or it and        repeat from step 5.    -   9. If the criterion is satisfied, check that path satisfies mass        deposition criterion.    -   10. If the mass deposition criterion is not satisfied, increase        n and/or m and repeat from step 5.    -   11. If the mass deposition criterion is satisfied, use scan        angle, y, to generate robot path and download path to robot.    -   12. Stop.

If it is not possible to find a scan angle that satisfies theoptimisation criterion, then the scan velocity and/or the width of thespray footprint need to be increased until the procedure can find asuitable path.

Theoretical derivation of optimised path plan according to the inventionis as follows.

Background: 2D Thermal Model

Partial Differential Equation

A 2D thermal model can be found using an energy balance for an elementof the steel shell.ΔE _(element) =E _(conducted) −E _(convected) +E _(supplied)  (1)where,

-   -   ΔE_(element)=increase in energy of element (J)    -   E_(conducted)=energy conducted into element (J)    -   E_(convected)=energy convected from element (J)    -   E_(supplied)=energy supplied by electric arc spray gun (J)        but,

$\begin{matrix}{{\Delta\; E_{element}} = {\rho\; c\;{z(t)}\frac{\partial\theta}{\partial t}\delta\; x\;\delta\; y}} & (2)\end{matrix}$where,

-   -   ρ=density of sprayed steel (kg m⁻³)    -   c=specific heat capacity of sprayed steel (J kg⁻¹ K⁻¹)    -   z(t)=thickness of steel shell (m)    -   θ(x, y, t)=temperature of element (K)    -   δxδy=area of element (m²)        and,        E _(conduction) =Kz(t)∇²θδxδy  (3)        where,    -   K=thermal conductivity of sprayed steel (W m⁻¹ K⁻¹)        and

$\begin{matrix}{{\nabla^{2}\theta} = {\frac{\partial^{2}\theta}{\partial x^{2}} + \frac{\partial^{2}\theta}{\partial y^{2}}}} & (4)\end{matrix}$and,E _(convection) =[H _(a)(θ−θ_(a))+H _(c)(θ−θ_(c))]δxδy  (5)where,

-   -   H_(a)=heat transfer coefficient from steel to air (W m⁻² K⁻¹)    -   θ_(a)=temperature of air (K)    -   H_(c)=heat transfer coefficient from steel to ceramic (W m⁻²        K⁻¹)    -   θ_(c)=temperature of ceramic (K)        and,        E _(supplied) =f(x, y, t)u(t)δxδy  (6)        where,    -   f(x, y, t)=thermal footprint of arc spray gun (J kg m⁻²)    -   υ(t)=wire feed rate to gun (kg s⁻¹)

Substituting (2), (3), (5) and (6) into (1) gives

$\begin{matrix}{{\rho\; c\;{z(t)}\frac{\partial\theta}{\partial t}\delta\; x\;\delta\; y} = {{K\;{z(t)}{\nabla^{2}{\theta\delta}}\; x\;\delta\; y} - {\left( {{H_{a}\left\lbrack {\theta - \theta_{a}} \right\rbrack} + {H_{c}\left\lbrack {\theta - \theta_{c}} \right\rbrack}} \right)\delta\; x\;\delta\; y} + {{f\left( {x,y,t} \right)}{u(t)}\delta\; x\;\delta\; y}}} & (7)\end{matrix}$and dividing through by pcz(t)δxδy leads to

$\begin{matrix}{\frac{\partial\theta}{\partial t} = {{\kappa{\nabla^{2}\theta}} - {{H(t)}\theta} + {{\overset{\sim}{f}\left( {x,y,t} \right)}{u(t)}} + {p(t)}}} & (8)\end{matrix}$where

$\kappa = \frac{K}{\rho\; c}$is the thermal diffusivity of sprayed steel and

$\begin{matrix}{{H(t)} = \frac{H_{a} + H_{c}}{\rho\; c\;{z(t)}}} & (9) \\{{\overset{\sim}{f}\left( {x,y,t} \right)} = \frac{f\left( {x,y,t} \right)}{\rho\; c\;{z(t)}}} & (10) \\{{p(t)} = \frac{{H_{a}\theta_{a}} + {H_{c}\theta_{c}}}{\rho\; c\;{z(t)}}} & (11)\end{matrix}$

The time dependence in the thermal footprint comes from the presence ofthe term z(t) in the denominator and from the movement of the gun overthe surface, so that

$\begin{matrix}{{f\left( {x,y,t} \right)} = \frac{\overset{\_}{f}\left( {{x - {\upsilon_{x}t}},{y - {\upsilon_{y}t}}} \right)}{\rho\; c\;{z(t)}}} & (12)\end{matrix}$where ν_(x) and ν_(y) are respectively, the robot velocity in the x andy directions and {tilde over (f)}(x′, y′) is the spray footprint, whichis independent of the position of the gun over the surface.Boundary Conditions

For a rectangular sheet of steel of length L_(x) and width L_(y) that isin contact with the air at the top and sides and underneath with thesurface of the ceramic, the heat loss from the top and bottom surfacesare modelled by the term H(t)θ+p(t) in (8). Provided that the sheet isthin, i.e. z(t)<<L_(x) and z(t)<<L_(y), it can be assumed that no heatis transferred from the sides of the sheet giving the Neumann boundaryconditions,

$\begin{matrix}{\frac{\partial\theta}{\partial t}{_{x = 0}{= {0\mspace{14mu}\frac{\partial\theta}{\partial x}{_{x = L_{x}}{= {0\mspace{14mu}\frac{\partial\theta}{\partial y}{_{y = 0}{= {0\mspace{14mu}\frac{\partial\theta}{\partial y}{_{y = L_{y}}{= 0}}}}}}}}}}}} & (13)\end{matrix}$together with the final value condition,θ(x, y, t)→0 as t→∞  (14)Solving the Partial Differential Equation

The aim is to solve (8) to find θ(x, y, t).

Homogeneous Part. Taking the homogeneous part of (8)

$\begin{matrix}{\frac{\partial\theta}{\partial t} = {{\kappa{\nabla^{2}\theta}} - {{H(t)}\theta}}} & (15)\end{matrix}$and assuming a separable solution of the form,θ(x, y, t)=q(t)φ(x, y)  (16)then upon substituting (16) into (15) gives,qφ=κ(qφ _(xx) +qφ _(yy))−H(t)qφ  (17)where,

$\begin{matrix}{{\phi_{x\; x} = {{\frac{\partial^{2}\phi}{\partial x^{2}}\mspace{31mu}\phi_{y\; y}} = \frac{\partial^{2}\phi}{\partial y^{2}}}}\mspace{11mu}} & (18)\end{matrix}$

Rearranging to get expressions in t on the left and expressions in (x,y) on the right, both sides can be set equal to a constant α giving,

$\begin{matrix}{{\frac{\overset{.}{q}}{\kappa\; q} + \frac{H(t)}{\kappa}} = {\frac{\left( {\phi_{x\; x} + \phi_{y\; y}} \right)}{\phi} = \alpha}} & (19)\end{matrix}$

This can be split into,q+|H(t)−q=0  (20)andφ_(xx)+φ_(yy)−αφ=0  (21)

Separating the solution for (21),φ(x, y)=X(x)Y(y)  (22)and substituting (22) into (21) gives,X″Y+XY″−αXY=0  (23)

Rearranging to get expressions in x on the left and expressions in y onthe right and setting both sides equal to a constant β, gives

$\begin{matrix}{\frac{X^{''}}{X} = {{{- \frac{Y^{''}}{Y}} + \alpha} = \beta}} & (24)\end{matrix}$which can be split into,X″−βX=0  (25)andY″−(α−β)Y=0  (26)

There are now three ordinary differential equations (ODE's), (20), (25)and (26), which can be solved. (20) is a homogeneous first order linearODE, whose solution isq(t)=A exp(−∫[H(t)−κα]dt)  (27)where A is a constant of integration. This satisfies the final valuecondition (14), provided that ∫[H(t)−κα]dt→∞ as t→∞.

Equation (25) is a second order ODE, which has a solution for β=−p²:X(x)=C cos px+D sin px  (28)

Applying the boundary conditions in (13) gives,

$\begin{matrix}{{{X_{m}(x)} = {C_{m}{\cos\left( {\frac{m\;\pi}{L_{x}}x} \right)}}}{{where},{p = \frac{m\;\pi}{L_{x}}}}} & (29)\end{matrix}$

Following a similar argument for (26) and writing (α−β)=−q², gives

$\begin{matrix}{{Y_{n}(y)} = {E_{n}{\cos\left( {\frac{n\;\pi}{L_{y}}y} \right)}}} & (30)\end{matrix}$where, q=nπ/L_(y).

Substituting (29) and (30) into (22) gives,

$\begin{matrix}{{\phi_{m,n}\left( {x,y} \right)} = {C_{m,n}^{\prime}{\cos\left( {\frac{m\;\pi}{L_{x}}x} \right)}{\cos\left( {\frac{n\;\pi}{L_{y}}y} \right)}}} & (31)\end{matrix}$where C_(m,n ′)=C_(m)E_(n). Since β=−p² and (α−β)=−q², then by definingλ_(m,n)=−α

$\begin{matrix}{\lambda_{m,n} = {\pi^{2}\left( {\frac{m^{2}}{L_{x}^{2}} + \frac{n^{2}}{L_{y}^{2}}} \right)}} & (32)\end{matrix}$

Combining (31), (27), (32) and (16) gives the full solution to thehomogeneous part of the PDE,

$\begin{matrix}{{\theta\left( {x,y,t} \right)} = {\sum\limits_{m = 0}^{\infty}\;{\sum\limits_{n = 0}^{\infty}\;{{q_{m,n}(t)}{\phi_{m,n}\left( {x,y} \right)}}}}} & (33) \\{{where},{{q_{m,n}(t)} = {F_{m,n}{\exp\left( {- {\int{\left\lbrack {{H(t)} + {\kappa\;\lambda_{m,n}}} \right){\mathbb{d}t}}}} \right)}}}} & (34) \\{{\phi_{m,n}\left( {x,y} \right)} = {{\cos\left( {\frac{m\;\pi}{L_{x}}x} \right)}{\cos\left( {\frac{n\;\pi}{L_{y}}y} \right)}}} & (35)\end{matrix}$with F_(m,n)=AC_(m,n)′.

Complete Solution Substituting equation (33) into equation (8) gives,

$\begin{matrix}{{\sum\limits_{m = 0}^{\infty}\;{\sum\limits_{n = 0}^{\infty}\;{{{\overset{.}{q}}_{m,n}(t)}{\phi_{m,n}\left( {x,y} \right)}}}} = {{\sum\limits_{m = 0}^{\infty}\;{\sum\limits_{n = 0}^{\infty}{{q_{m,n}(t)}\left\lbrack {{\kappa\frac{\partial^{2}\phi_{m,n}}{d\; x^{2}}} + {\kappa\frac{\partial^{2}\phi_{m,n}}{d\; y^{2}}} - {{H(t)}{\phi_{m,n}\left( {x,y} \right)}}} \right\rbrack}}} + {{\overset{\sim}{f}\left( {x,\overset{¨}{y},t} \right)}\overset{\sim}{u}(t)} + {p(t)}}} & (36) \\{{{{but}\mspace{31mu}\frac{\partial^{2}\phi_{m,n}}{\partial^{2}x}} + \frac{\partial^{2}\phi_{m,n}}{\partial^{2}y}} = {{- \lambda_{m,n}}\phi_{m,n}}} & (37)\end{matrix}$which gives,

$\begin{matrix}{{\sum\limits_{m = 0}^{\infty}\;{\sum\limits_{n = 0}^{\infty}\;{{{\overset{.}{q}}_{m,n}(t)}{\phi_{m,n}\left( {x,y} \right)}}}} = {{\sum\limits_{m = 0}^{\infty}\;{\sum\limits_{n = 0}^{\infty}{{- \left\lbrack {{H(t)} + {\kappa\;\lambda_{m,n}}} \right\rbrack}{q_{m,n}(t)}\phi_{m,n}}}} + {{\overset{\sim}{f}\left( {x,y,t} \right)}{u(t)}} + {p(t)}}} & (38)\end{matrix}$

Multiplying both sides by φ_(m′,n′)(x, y) and integrating gives,

$\begin{matrix}{{\sum\limits_{m = 0}^{\infty}\;{\sum\limits_{n = 0}^{\infty}{{\overset{.}{q}}_{m,n}{\int_{0}^{L_{y}}{\int_{0}^{L_{x}}{\phi_{m^{\prime},n^{\prime}}\phi_{m,n}\ {\mathbb{d}x}{\mathbb{d}y}}}}}}} = {{\sum\limits_{m = 0}^{\infty}\;{\sum\limits_{n = 0}^{\infty}{{- \left\lbrack {{H(t)} + {\kappa\;\lambda_{m,n}}} \right\rbrack}q_{m,n}{\int_{0}^{L_{y}}{\int_{0}^{L_{x}}{\phi_{m^{\prime},n^{\prime}}\phi_{m,n}\ {\mathbb{d}x}{\mathbb{d}y}}}}}}} + {{u(t)}{\int_{0}^{L_{y}}{\int_{0}^{L_{x}}{\phi_{m^{\prime},n^{\prime}}{\overset{\sim}{f}\left( {x,y,t} \right)}{\mathbb{d}x}{\mathbb{d}y}}}}} + {{p(t)}{\int_{0}^{L_{y}}{\int_{0}^{L_{x}}{\phi_{m^{\prime},n^{\prime}}{\mathbb{d}x}{\mathbb{d}y}}}}}}} & (39)\end{matrix}$

Using the orthogonality of φ_(m,n)(x, y)

$\begin{matrix}{{\int_{0}^{L_{y}}{\int_{0}^{L_{x}}{\phi_{m^{\prime},n^{\prime}}\phi_{m,n}\ {\mathbb{d}x}{\mathbb{d}y}}}} = \left\{ \begin{matrix}\frac{L_{x}L_{y}}{4} & {for} & {m = {{m^{\prime}\mspace{14mu}{and}\mspace{14mu} n} = n^{\prime}}} \\0 & {for} & {m \neq {m^{\prime}\mspace{14mu}{or}\mspace{14mu} n} \neq n^{\prime}}\end{matrix} \right.} & (40)\end{matrix}$together with

$\begin{matrix}{{\int_{0}^{L_{y}}{\int_{0}^{L_{x}}{\phi_{m^{\prime},n^{\prime}}\ {\mathbb{d}x}{\mathbb{d}y}}}} = \left\{ \begin{matrix}{L_{x}L_{y}} & {for} & {m^{\prime} = {{0\mspace{14mu}{and}\mspace{14mu} n^{\prime}} = 0}} \\0 & {for} & {m^{\prime} \neq {0\mspace{14mu}{or}\mspace{14mu} n^{\prime}} \neq 0}\end{matrix} \right.} & (41)\end{matrix}$reduces (39) to

$\begin{matrix}{{\overset{.}{q}}_{m,n} = {{{- \left\lbrack {{H(t)} + {\kappa\;\lambda_{m,n}}} \right\rbrack}q_{m,n}} + {{u(t)}\frac{4}{L_{x}L_{y}}{\int_{0}^{L_{y}}{\int_{0}^{L_{x}}{\phi_{m,n}\ {\overset{\sim}{f}\left( {x,y,t} \right)}{\mathbb{d}x}{\mathbb{d}y}}}}}}} & (42)\end{matrix}$

Remark. There will be an additional term, p(t)L_(x)L_(y), that is addedto the expression for q_(0,0), but because this only affects the DCmode, this term will be ignored.

If b_(m,n)(t) is used to denote

$\begin{matrix}{{b_{m,n}(t)} = {\frac{4}{L_{x}L_{y}}{\int_{0}^{L_{y}}{\int_{0}^{L_{x}}\ {{\overset{\sim}{f}\left( {x,y,t} \right)}\phi_{m,n}{\mathbb{d}x}{\mathbb{d}y}}}}}} & (43)\end{matrix}$then rearranging (42) gives,{dot over (q)} _(m,n)(t)=−[H(t)+κλ_(m,n) ]q _(m,n)(t)+b_(m,n)(t)u(t)  (44)for each thermal mode.Impulsive Heat Source

Consider the case of a rectangular surface of length L_(x) and widthL_(y), insulated on all sides, which is healed by an impulse heat sourcemoving with velocity ν at an angle ψ to the side of the rectangle. Thevelocity of the source call be split into its components in the x and ydirections giving,ν_(x)=ν cos ψt  (45)and,ν_(y)=ν sin ψt  (46)

If the spatial profile of the heat source is a delta function, then (towithin a scaling factor){tilde over (f)}(x, y, t)=δ(x−ν_(x) t, y−ν _(y) t)  (47)

Each of the states associated with the spatial modes satisfies (44)where,

$\begin{matrix}\begin{matrix}{{{b_{m,n}(t)} = {\frac{4}{L_{x}L_{y}}{\int_{0}^{L_{y}}{\int_{0}^{L_{x}}{\delta\left( {{x - {\upsilon_{x}t}},{y - {\upsilon_{y}t}}} \right)}}}}}\ } \\{{\cos\left( \frac{m\;\pi\; x}{L_{x}} \right)}{\cos\left( \frac{n\;\pi\; y}{L_{y}} \right)}{\mathbb{d}x}{\mathbb{d}y}}\end{matrix} & (48) \\{\mspace{65mu}{= {\frac{4}{L_{x}L_{y}}{\cos\left( {\frac{m\;\pi\;\upsilon_{x}}{L_{x}}t} \right)}{\cos\left( {\frac{n\;\pi\;\upsilon_{y}}{L_{y}}t} \right)}}}} & (49)\end{matrix}$

Because the system is assumed to have Neumann boundary conditions thespatial eigenfunctions, φ_(m,n)(x, y) consist of cosine functions. As aresult, φ_(m,n)(x, y) is an even function with respect to both x and y,so this expression for b_(m,n)(t) holds as the sign of ν_(x) and ν_(y)switches when the heat source changes direction at the edge of thesurface.

The expression for b_(m,n)(t) in (49) can be rearranged using atrigonometric identity to give,

$\begin{matrix}{{b_{m,n}(t)} = {\frac{2}{L_{x}L_{y}}\left\lbrack {{{\cos\left( {\frac{m\;\pi\;\upsilon_{x}}{L_{x}} + \frac{n\;\pi\;\upsilon_{y}}{L_{y}}} \right)}t} + {{\cos\left( {\frac{m\;\pi\;\upsilon_{x}}{L_{x}} - \frac{n\;\pi\;\upsilon_{y}}{L_{y}}} \right)}t}} \right\rbrack}} & (50) \\{\mspace{70mu}{= {\frac{2}{L_{x}L_{y}}\left\lbrack {{\cos\;\omega_{1}t} + {\cos\;\omega_{2}t}} \right\rbrack}}} & (51) \\{{where},{\omega_{1} = {{\frac{m\;\pi\;\upsilon_{x}}{L_{x}} + {\frac{n\;\pi\;\upsilon_{y}}{L_{y}}\mspace{25mu}\omega_{2}}} = {\frac{m\;\pi\;\upsilon_{x}}{L_{x}} - \frac{n\;\pi\;\upsilon_{y}}{L_{y}}}}}} & (52)\end{matrix}$

The term H(t) is time-varying due to the change in z(t), the meanthickness of the steel. Since the thickness builds up slowly, it isreasonable to assume that H(t) will remain approximately constant overthe period of a complete cycle of scans. If the wire feed rate is alsoconstant, so that υ(t)=υ₀, then applying Laplace transforms to (44)(assuming that q_(m,n)(0)=0) gives,sQ _(m,n)(s)=−[H(t)+κλ_(m,n) ]Q _(m,n)(s)+B _(m,n)(s)υ ₀  (53)which leads to an expression-for G(s), the transfer function fromB_(m,n)(s) to Q_(m,n)(s)

$\begin{matrix}{\frac{Q_{m,n}(s)}{B_{m,n}(s)} = {{G(s)} = \frac{u_{0}}{s + {H(t)} + {\kappa\;\lambda_{m,n}}}}} & (54)\end{matrix}$so that

$\begin{matrix}{{{q_{m,n}(t)} = {{{{G\left( {j\;\omega_{1}} \right)}}\frac{2}{L_{x}L_{y}}{\cos\left\lbrack {{\omega_{1}t} + {\angle\;{G\left( {j\;\omega_{1}} \right)}}} \right\rbrack}} + {{{G\left( {j\;\omega_{2}} \right)}}\frac{2}{L_{x}L_{y}}{\cos\left\lbrack {{\omega_{2}t} + {\angle\;{G\left( {j\;\omega_{2}} \right)}}} \right\rbrack}}}}{where}} & (55) \\{{{G\left( {j\;\omega} \right)}} = \frac{u_{0}}{\sqrt{\omega^{2} + \left\lbrack {{H(t)} + {\kappa\lambda}_{m,n}} \right\rbrack^{2}}}} & (56) \\{{\angle\;{G\left( {j\;\omega} \right)}} = {- {\arctan\left( \frac{\omega}{{H(t)} + {\kappa\lambda}_{m,n}} \right)}}} & (57)\end{matrix}$giving

$\begin{matrix}{{q_{m,n}(t)} = {{\frac{2}{L_{x}L_{v}}\frac{u_{0}}{\sqrt{\left( {\frac{m\;{\pi\upsilon}_{x}}{L_{x}} + \frac{n\;{\pi\upsilon}_{y}}{L_{y}}} \right)^{2} + \left\lbrack {{H(t)} + {\kappa\lambda}_{m,n}} \right\rbrack^{2}}}{\cos\left\lbrack {{\left( {\frac{m\;{\pi\upsilon}_{x}}{L_{x}} + \frac{n\;{\pi\upsilon}_{y}}{L_{y}}} \right)t} - {\arctan\left( \frac{{m\;{\pi\upsilon}_{x}L_{y}} + {n\;{\pi\upsilon}_{y}L_{x}}}{{{H(t)}L_{x}L_{y}} + {{\kappa\lambda}_{m,n}L_{x}L_{y}}} \right)}} \right\rbrack}} + {\frac{2}{L_{x}L_{y}}\frac{u_{0}}{\sqrt{\left( {\frac{m\;{\pi\upsilon}_{x}}{L_{x}} - \frac{n\;{\pi\upsilon}_{y}}{L_{y}}} \right)^{2} + \left\lbrack {{H(t)} + {\kappa\lambda}_{m,n}} \right\rbrack^{2}}}{\cos\left\lbrack {{\left( {\frac{m\;{\pi\upsilon}_{z}}{L_{x}} - \frac{n\;{\pi\upsilon}_{y}}{L_{y}}} \right)t} - {\arctan\left( \frac{{m\;{\pi\upsilon}_{x}L_{y}} - {n\;{\pi\upsilon}_{y}L_{x}}}{{{H(t)}L_{x}L_{y}} + {{\kappa\lambda}_{m,n}L_{x}L_{y}}} \right)}} \right\rbrack}}}} & (58)\end{matrix}$General Heat Sources

The analysis above assumes that the surface is heated by a source whichhas a spatial profile consisting of an impulse function, δ(x, y). Thisis a specific case of the more general 2D heat source {tilde over(f)}(x, y, t). In the general case, with the heat source moving withvelocity ν and angle ψ over the surface,

$\begin{matrix}{\frac{\partial\theta}{\partial t} = {{\kappa{\nabla^{2}\theta}} - {{H(t)}\theta} + {{\overset{\sim}{f}\left( {{x - {\upsilon_{x}t}},{y - {\upsilon_{y}t}}} \right)}{u(t)}} + {p(t)}}} & (59)\end{matrix}$and the coefficients, q_(m,n)(t), have the solution{dot over (q)} _(m,n)(t)=−[H(t)+κλ_(m,n)](t)+{tilde over (b)} _(m,n)(t)υ₀  (60)where,

$\begin{matrix}{{{\overset{\sim}{b}}_{m,n}(t)} = {\frac{4}{L_{x}L_{y}}{\int_{0}^{L_{y}}{\int_{0}^{L_{x}}{{\overset{\sim}{f}\left( {{x - {\upsilon_{x}t}},{y - {\upsilon_{y}t}}} \right)}{\cos\left( \frac{m\;\pi\; x}{L_{x}} \right)}{\cos\ \left( \frac{n\;\pi\; y}{L_{y}} \right)}{\mathbb{d}x}{\mathbb{d}y}}}}}} & (61)\end{matrix}$

Applying a change of variables, x′=x−ν_(x)t, y′=ν_(y)t

$\begin{matrix}{{{\overset{\sim}{b}}_{m,n}(t)} = {\frac{4}{L_{x}L_{y}}{\int_{{- \upsilon_{y}}t}^{L_{y} - {\upsilon_{y}t}}{\int_{{- \upsilon_{x}}t}^{L_{x} - {\upsilon_{x}t}}\ {{\overset{\sim}{f}\left( {x^{\prime},y^{\prime}} \right)}{\cos\left( \frac{m\;\pi\;\left( {x^{\prime} + {\upsilon_{x}t}} \right)}{L_{x}} \right)}{\cos\ \left( \frac{n\;\pi\;\left( {y^{\prime} + {\upsilon_{y}t}} \right)}{L_{y}} \right)}{\mathbb{d}x^{\prime}}{\mathbb{d}y^{\prime}}}}}}} & (62)\end{matrix}$

Using cos(A+B)=cos A cos B−sin A sin B

$\begin{matrix}{{{\overset{\sim}{b}}_{m,n}(t)} = {{\frac{4}{L_{x}L_{y}}{\cos\left( \frac{m\;{\pi\upsilon}_{x}t}{L_{x}} \right)}{\cos\left( \frac{n\;{\pi\upsilon}_{y}t}{L_{y}} \right)}{\int_{{- \upsilon_{y}}t}^{L_{y} - {\upsilon_{y}t}}{\int_{{- \upsilon_{x}}t}^{L_{x} - {\upsilon_{x}t}}\ {{\overset{\sim}{f}\left( {x^{\prime},y^{\prime}} \right)}{\cos\left( \frac{m\;\pi\; x^{\prime}}{L_{x}} \right)}{\cos\ \left( \frac{n\;\pi\; y^{\prime}}{L_{y}} \right)}{\mathbb{d}x^{\prime}}{\mathbb{d}y^{\prime}}}}}} - {\frac{4}{L_{x}L_{y}}{\cos\left( \frac{m\;{\pi\upsilon}_{x}t}{L_{x}} \right)}{\sin\left( \frac{n\;{\pi\upsilon}_{y}t}{L_{y}} \right)}{\int_{{- \upsilon_{y}}t}^{L_{y} - {\upsilon_{y}t}}{\int_{{- \upsilon_{x}}t}^{L_{x} - {\upsilon_{x}t}}\ {{\overset{\sim}{f}\left( {x^{\prime},y^{\prime}} \right)}{\cos\left( \frac{m\;\pi\; x^{\prime}}{L_{x}} \right)}{\sin\ \left( \frac{n\;\pi\; y^{\prime}}{L_{y}} \right)}{\mathbb{d}x^{\prime}}{\mathbb{d}y^{\prime}}}}}} - {\frac{4}{L_{x}L_{y}}{\sin\left( \frac{m\;{\pi\upsilon}_{x}t}{L_{x}} \right)}{\cos\left( \frac{n\;{\pi\upsilon}_{y}t}{L_{y}} \right)}{\int_{{- \upsilon_{y}}t}^{L_{y} - {\upsilon_{y}t}}{\int_{{- \upsilon_{x}}t}^{L_{x} - {\upsilon_{x}t}}\ {{\overset{\sim}{f}\left( {x^{\prime},y^{\prime}} \right)}{\sin\left( \frac{m\;\pi\; x^{\prime}}{L_{x}} \right)}{\cos\ \left( \frac{n\;\pi\; y^{\prime}}{L_{y}} \right)}{\mathbb{d}x^{\prime}}{\mathbb{d}y^{\prime}}}}}} + {\frac{4}{L_{x}L_{y}}{\sin\left( \frac{m\;{\pi\upsilon}_{x}t}{L_{x}} \right)}{\sin\left( \frac{n\;{\pi\upsilon}_{y}t}{L_{y}} \right)}{\int_{{- \upsilon_{y}}t}^{L_{y} - {\upsilon_{y}t}}{\int_{{- \upsilon_{x}}t}^{L_{x} - {\upsilon_{x}t}}\ {{\overset{\sim}{f}\left( {x^{\prime},y^{\prime}} \right)}{\sin\left( \frac{m\;\pi\; x^{\prime}}{L_{x}} \right)}{\sin\ \left( \frac{n\;\pi\; y^{\prime}}{L_{y}} \right)}{\mathbb{d}x^{\prime}}{\mathbb{d}y^{\prime}}}}}}}} & (63)\end{matrix}$

If the spatial range of the heat source is limited, so that {tilde over(f)}(x′, y′)=0 for |x′|>γ_(x) and for |y′|>γ_(y), then the limits of theintegrations in (63) can be truncated. In addition, if {tilde over(f)}(x′, y′) is an even function with respect to both x′ and y′, thenonly the integrand in the first integral is also even. The other threeintegrands are odd functions and will therefore integrate to zeroprovided that −ν_(x)t≦−γ_(x), L_(x)−ν_(x)t≧γ_(x), −ν_(y)t≦−γ_(y) andL_(y)−ν_(y)t≧γ_(y). Clearly this will not be the case when the heatsource is close to the edges of the surface, so an error will beintroduced. If L_(x)>>γ_(x) and L_(y)>>γ_(y) this error will be smalland will be ignored in the rest of the analysis.

Equation (63) reduces to

$\begin{matrix}{{{\overset{\sim}{b}}_{m,n}(t)} = {{\overset{\sim}{b}}_{m,n}\frac{4}{L_{x}L_{y}}{\cos\left( \frac{m\;{\pi\upsilon}_{x}t}{L_{x}} \right)}{\cos\left( \frac{n\;{\pi\upsilon}_{y}t}{L_{y}} \right)}}} & (64)\end{matrix}$where {circumflex over (b)}_(m,n) is obtained from the expansion of thespatial footprint of the gun when it is positioned in the centre of thesurface, so that the region where {tilde over (f)}(x, y, t)≠0 does notextend beyond the edges of the surface. The gun will be at the centre ofthe surface when ν_(x)t=L_(x)/2 and ν_(y)t=L_(y)/2, so that

$\begin{matrix}{{\overset{\sim}{b}}_{m,n} = {\int_{{- L_{y}}/2}^{L_{y}/2}{\int_{{- L_{x}}/2}^{L_{x}/2}\ {{\overset{\sim}{f}\left( {x^{\prime},y^{\prime}} \right)}{\cos\left( \frac{m\;\pi\; x^{\prime}}{L_{x}} \right)}{\cos\ \left( \frac{n\;\pi\; y^{\prime}}{L_{y}} \right)}{\mathbb{d}x^{\prime}}{\mathbb{d}y^{\prime}}}}}} & (65)\end{matrix}$

This shows that for a general heat source, {tilde over (f)}(x, y, t),the coefficients associated with each mode, q_(m,n)(t) for a generalheat source reduce to the solution for an impulsive heat sourcemultiplied by {circumflex over (b)}_(m,n)

$\begin{matrix}{{{q_{m,n}(t)} = {{\frac{2}{L_{x}L_{v}}\frac{{\overset{\sim}{b}}_{m,n}u_{0}}{\sqrt{\left( {\frac{m\;{\pi\upsilon}_{x}}{L_{x}} + \frac{n\;{\pi\upsilon}_{y}}{L_{y}}} \right)^{2} + \left\lbrack {{H(t)} + {\kappa\lambda}_{m,n}} \right\rbrack^{2}}}{\cos\left\lbrack {{\left( {\frac{m\;{\pi\upsilon}_{x}}{L_{x}} + \frac{n\;{\pi\upsilon}_{y}}{L_{y}}} \right)t} - {\arctan\left( \frac{{m\;{\pi\upsilon}_{x}L_{y}} + {n\;{\pi\upsilon}_{y}L_{x}}}{{{H(t)}L_{x}L_{y}} + {{\kappa\lambda}_{m,n}L_{x}L_{y}}} \right)}} \right\rbrack}} + {\frac{2}{L_{x}L_{y}}\frac{{\overset{\sim}{b}}_{m,n}u_{0}}{\sqrt{\left( {\frac{m\;{\pi\upsilon}_{x}}{L_{x}} - \frac{n\;{\pi\upsilon}_{y}}{L_{y}}} \right)^{2} + \left\lbrack {{H(t)} + {\kappa\lambda}_{m,n}} \right\rbrack^{2}}}{\cos\left\lbrack {{\left( {\frac{m\;{\pi\upsilon}_{z}}{L_{x}} - \frac{n\;{\pi\upsilon}_{y}}{L_{y}}} \right)t} - {\arctan\left( \frac{{m\;{\pi\upsilon}_{x}L_{y}} - {n\;{\pi\upsilon}_{y}L_{x}}}{{{H(t)}L_{x}L_{y}} + {{\kappa\lambda}_{m,n}L_{x}L_{y}}} \right)}} \right\rbrack}}}}{where}} & (66) \\{\lambda_{m,n} = {\frac{m^{2}\pi^{2}}{L_{x}^{2}} + \frac{n^{2}\pi^{2}}{L_{y}^{2}}}} & (67)\end{matrix}$Choosing the Optimal PathOptimisation Criteria

The thermal profile over the surface is

$\begin{matrix}{{\theta\left( {x,y,t} \right)} = {\sum\limits_{m = 0}^{\infty}\;{\sum\limits_{n = 0}^{\infty}{{q_{m,n}(t)}{\phi_{m,n}\left( {x,y} \right)}}}}} & (68)\end{matrix}$where q_(m,n)(t) are given in (66) and from (35), the spatialeigenfunctions, are

$\begin{matrix}{{\phi_{m,n}\left( {x,y} \right)} = {{\cos\left( {\frac{{m\;\pi}\;}{L_{x}}x} \right)}{\cos\ \left( {\frac{{n\;\pi}\;}{L_{y}}y} \right)}}} & (69)\end{matrix}$

The deviation from the average temperature is given byθ(x, y, t)−θ(t)  (70)where

$\begin{matrix}{{\overset{\sim}{\theta}(t)} = {\frac{1}{L_{x}L_{y}}{\int_{0}^{L_{y}}{\int_{0}^{L_{y}}{{\theta\left( {x,y,t} \right)}{\mathbb{d}x}{\mathbb{d}y}}}}}} & (71) \\{\mspace{34mu}{= {\sum\limits_{m = 0}^{\infty}\;{\sum\limits_{n = 0}^{\infty}{{q_{m,n}(t)}\frac{1}{L_{x}L_{y}}{\int_{0}^{L_{y}}{\int_{0}^{L_{y}}{{\cos\left( {\frac{{m\;\pi}\;}{L_{x}}x} \right)}{\cos\ \left( {\frac{{n\;\pi}\;}{L_{y}}y} \right)}{\mathbb{d}x}{\mathbb{d}y}}}}}}}}} & (72) \\{\mspace{31mu}{= {q_{0,0}(t)}}} & (73)\end{matrix}$since φ_(0,0)(x, y)=1. Hence, the deviation in the temperature profileis obtained by removing the 0, 0 term from the summations in (68)

$\begin{matrix}{{{\theta\left( {x,y,t} \right)} - {\overset{\sim}{\theta}(t)}} = {\sum\limits_{m = 1}^{\infty}\;{\sum\limits_{n = 1}^{\infty}{{q_{m,n}(t)}{\phi_{m,n}\left( {x,y} \right)}}}}} & (74)\end{matrix}$

This justifies excluding the p(t) term in (42) as it only affects theq_(0,0)(t) term which does not contribute to the deviation from theaverage temperature.

The aim is to choose a path for the spray gun that minimises (in somesense), the deviation. There are a number of approaches to minimisingthe deviation in temperature, but three appropriate choices areconsidered here Maximum Deviation At any time, t, the maximum value ofthe temperature deviation over the surface is given by

$\begin{matrix}{{{{\theta\left( {x,y,t} \right)} - {\overset{\sim}{\theta}(t)}}} = {{\sum\limits_{m = 1}^{\infty}\;{\sum\limits_{n = 1}^{\infty}{{q_{m,n}(t)}{\phi_{m,n}\left( {x,y} \right)}}}}}} & (75) \\{\mspace{169mu}{\leq {\sum\limits_{m = 1}^{\infty}\;{\sum\limits_{n = 1}^{\infty}{{{q_{m,n}(t)}{\phi_{m,n}\left( {x,y} \right)}}}}}}} & (76) \\{\mspace{166mu}{\leq {\sum\limits_{m = 1}^{\infty}\;{\sum\limits_{n = 1}^{\infty}{{q_{m,n}(t)}}}}}} & (77)\end{matrix}$where (77) follows because the maximum value of φ_(m,n)(x, y) over thesurface is unity for all spatial eigenvalues. The peak value of|q_(m,n)(t)| will occur at times when the two cosine components in (66)interfere constructively, so that

$\begin{matrix}{{q_{m,n}}_{peak} = {\frac{2}{L_{x}L_{y}}\left\lbrack {\frac{{\hat{b}}_{m,n}u_{0}}{\sqrt{\left( {\frac{m\;{\pi\upsilon}_{x}}{L_{x}} + \frac{n\;{\pi\upsilon}_{y}}{L_{y}}} \right)^{2} + \left\lbrack {{H(t)} + {\kappa\lambda}_{m,n}} \right\rbrack^{2}}} + \frac{{\hat{b}}_{m,n}u_{0}}{\sqrt{\left( {\frac{m\;{\pi\upsilon}_{x}}{L_{x}} - \frac{n\;{\pi\upsilon}_{y}}{L_{y}}} \right)^{2} + \left\lbrack {{H(t)} + {\kappa\lambda}_{m,n}} \right\rbrack^{2}}}} \right\rbrack}} & (78)\end{matrix}$

The overall deviation in temperature is minimsed by minimising themaximum peak value, |q_(m,n)|_(peak), for {m=1,2, . . . ,n=1,2, . . . }.

Thermal Gradient The gradient of the temperature deviation in thex-direction is

$\begin{matrix}{\frac{\partial\theta}{\partial x} = {\sum\limits_{m = 1}^{\infty}\;{\sum\limits_{n = 1}^{\infty}{{q_{m,n}(t)}\frac{\partial\phi_{m,n}}{\partial x}}}}} & (79) \\{\mspace{34mu}{= {- {\sum\limits_{m = 1}^{\infty}\;{\sum\limits_{n = 1}^{\infty}{\frac{m\;\pi}{L_{x}}{q_{m,n}(t)}{\phi_{m,n}\left( {x,y} \right)}}}}}}} & (80)\end{matrix}$

Hence the magnitude of the thermal gradient in the x-direction can beminimised by minimising the maximum value of

$\begin{matrix}{{{\frac{m\;\pi}{L_{x}}q_{m,n}}}_{peak} = {\frac{m\;\pi}{L_{x}}{\frac{2}{L_{x}L_{y}}\left\lbrack {\frac{{\hat{b}}_{m,n}u_{0}}{\sqrt{\left( {\frac{m\;{\pi\upsilon}_{x}}{L_{x}} + \frac{n\;{\pi\upsilon}_{y}}{L_{y}}} \right)^{2} + \left\lbrack {{H(t)} + {\kappa\lambda}_{m,n}} \right\rbrack^{2}}} + \frac{{\hat{b}}_{m,n}u_{0}}{\sqrt{\left( {\frac{m\;{\pi\upsilon}_{x}}{L_{x}} - \frac{n\;{\pi\upsilon}_{y}}{L_{y}}} \right)^{2} + \left\lbrack {{H(t)} + {\kappa\lambda}_{m,n}} \right\rbrack^{2}}}} \right\rbrack}}} & (81)\end{matrix}$

This criterion is similar to minimising the deviation, but more “weight”is applied to the magnitude of the higher order modes, which generatelarger thermal gradients. The magnitude of the thermal gradient in they-direction is minimised by minimising the maximum value of

$\begin{matrix}{{{\frac{m\;\pi}{L_{x}}q_{m,n}}}_{peak} = {\frac{n\;\pi}{L_{y}}{\frac{2}{L_{x}L_{y}}\left\lbrack {\frac{{\hat{b}}_{m,n}u_{0}}{\sqrt{\left( {\frac{m\;{\pi\upsilon}_{x}}{L_{x}} + \frac{n\;{\pi\upsilon}_{y}}{L_{y}}} \right)^{2} + \left\lbrack {{H(t)} + {\kappa\lambda}_{m,n}} \right\rbrack^{2}}} + \frac{{\hat{b}}_{m,n}u_{0}}{\sqrt{\left( {\frac{m\;{\pi\upsilon}_{x}}{L_{x}} - \frac{n\;{\pi\upsilon}_{y}}{L_{y}}} \right)^{2} + \left\lbrack {{H(t)} + {\kappa\lambda}_{m,n}} \right\rbrack^{2}}}} \right\rbrack}}} & (82)\end{matrix}$

Mean Square Deviation The mean square deviation (or variance) of thethermal profile over the surface is

$\begin{matrix}{{\frac{1}{L_{x}L_{y}}{\int_{0}^{L_{y}}{\int_{0}^{L_{x}}{\left\lbrack {{\theta\left( {x,y,t} \right)} - {\overset{\sim}{\theta}(t)}} \right\rbrack^{2}{\mathbb{d}x}{\mathbb{d}y}}}}} = {\frac{1}{L_{x}L_{y}}{\int_{0}^{L_{y}}{\int_{0}^{L_{x}}{\left\lbrack {\sum\limits_{m = 1}^{\infty}\;{\sum\limits_{n = 1}^{\infty}{{q_{m,n}(t)}{\phi_{m,n}\left( {x,y} \right)}}}} \right\rbrack^{2}{\mathbb{d}x}{\mathbb{d}y}}}}}} & (83)\end{matrix}$

By the orthogonality properties of the spatial eigenfunctions,φ_(m,n)(x, y), this reduces to

$\begin{matrix}{{\frac{1}{L_{x}L_{y}}{\int_{0}^{L_{y}}{\int_{0}^{L_{x}}{\left\lbrack {{\theta\left( {x,y,t} \right)} - {\overset{\sim}{\theta}(t)}} \right\rbrack^{2}{\mathbb{d}x}{\mathbb{d}y}}}}} = {\sum\limits_{m = 1}^{\infty}\;{\sum\limits_{n = 1}^{\infty}\left\lbrack {q_{m,n}(t)} \right\rbrack^{2}}}} & (84)\end{matrix}$where q_(m,n)(t) is given in (66). It this mean square deviation isaveraged over time,

$\begin{matrix}{{\lim\limits_{T->\infty}{\frac{1}{T}{\int_{0}^{T}{\sum\limits_{m = 1}^{\infty}\;{\sum\limits_{n = 1}^{\infty}\ {\left. {q_{m,n}(t)} \right\rbrack^{2}{\mathbb{d}t}}}}}}} = {{\sum\limits_{m = 1}^{\infty}\;{\sum\limits_{n = 1}^{\infty}{\lim\limits_{t->\infty}{\frac{1}{T}{\int_{0}^{T}{\left. {q_{m,n}(t)} \right\rbrack^{2}{\mathbb{d}t}}}}}}} =}} & (85) \\{\mspace{101mu}{\sum\limits_{m = 1}^{\infty}\;{\sum\limits_{n = 1}^{\infty}{\frac{1}{L_{x}L_{y}}\;\left\lbrack {\frac{{\hat{b}}_{m,n}^{2}u_{0}^{2}}{\left( {\frac{m\;{\pi\upsilon}_{x}}{L_{x}} + \frac{n\;{\pi\upsilon}_{y}}{L_{y}}} \right)^{2} + \left\lbrack {{H(t)} + {\kappa\lambda}_{m,n}} \right\rbrack^{2}} + \frac{{\hat{b}}_{m,n}^{2}u_{0}^{2}}{\left( {\frac{m\;{\pi\upsilon}_{x}}{L_{x}} - \frac{n\;{\pi\upsilon}_{y}}{L_{y}}} \right)^{2} + \left\lbrack {{H(t)} + {\kappa\lambda}_{m,n}} \right\rbrack^{2}}} \right\rbrack}}}} & (86)\end{matrix}$Effect of Changing Path on Thermal Profile

For each of the criteria listed above, the magnitude of the criteria aredetermined by the maximum amplitude of the oscillations in q_(m,n)(t)for each mode

$\begin{matrix}{{q_{m,n}}_{peak} = {\frac{2}{L_{x}L_{y}}\left\lbrack {\frac{{\hat{b}}_{m,n}u_{0}}{\sqrt{\left( {\frac{m\;{\pi\upsilon}_{x}}{L_{x}} + \frac{n\;{\pi\upsilon}_{y}}{L_{y}}} \right)^{2} + \left\lbrack {{H(t)} + {\kappa\lambda}_{m,n}} \right\rbrack^{2}}} + \frac{{\hat{b}}_{m,n}u_{0}}{\sqrt{\left( {\frac{m\;{\pi\upsilon}_{x}}{L_{x}} - \frac{n\;{\pi\upsilon}_{y}}{L_{y}}} \right)^{2} + \left\lbrack {{H(t)} + {\kappa\lambda}_{m,n}} \right\rbrack^{2}}}} \right\rbrack}} & (87)\end{matrix}$

When choosing a regular scanning path, there are two degrees of freedomfor adjusting the magnitude of each mode:

-   Scan velocity, ν. Prom equation (87), it can be seen that increasing    the velocity, ν, reduces the magnitude of all modes and there is an    approximately inverse relationship between the amplitude of    q_(m,n)(t) and scan velocity, as shown in FIG. 3, which plots the    amplitude of different spatial modes against scan velocity, ν. As a    result, to achieve a “flat” temperature profile, the scan velocity    should be as fast as possible.-   Scan angle, ψ. The relationship between the amplitude of q_(m,n)(t)    and the scan angle is more complicated. The first term inside the    square brackets is large when m and n, and consequently, λ_(m,n),    are small. The second term is maximised when

$\begin{matrix}{{\frac{m\;{\pi\upsilon}\;\cos\;\psi}{L_{x}} + \frac{n\;{\pi\upsilon sin\psi}}{L_{y}}}{or}} & (88) \\{{\tan\;\psi} = \frac{m\; L_{y}}{n\; L_{x}}} & (89)\end{matrix}$so that the first term under the square root in the denominator becomeszero. This is illustrated in FIG. 4 which shows the relative magnitudeof the different modes when {tilde over (f)}(x, y, t) is a 2-dimensionalGaussian function with circular symmetry of width L_(x)/20. For a squaresurface, so that L_(x)=L_(y), when the scan angle is 56.13°, tan ψ=3/2and the plot shows that the q_(3,2)(t) mode has the maximum amplitude.As a result, this scanning pattern will result in a poor thermalprofile. By contrast, for the path shown in FIG. 5, ψ=72.9°, so that tanψ=3.25, then the relative magnitude of the modes is much lower. The modewith the largest amplitude on the plot is q_(3,1)(t) as this is closestto tan ψ. Since tan ψ=13/4, it might be expected that q_(13,4)(t) wouldhave the largest amplitude, but the presence of the m² and n² terms in

$\begin{matrix}{\lambda_{m,n} = {\pi^{2}{\upsilon^{2}\left( {\frac{m^{2}\cos^{2}\psi}{L_{x}^{2}} + \frac{n^{2}\sin^{2}\psi}{L_{y}^{2}}} \right)}}} & (90)\end{matrix}$which is also under the square root in the denominator, reduces theamplitude of this mode. In addition, if {tilde over (f)}(x, y, t) issmooth, so that | b _(m,n)|→0 as m and n become large and b _(13,4) islikely to be small.

The effect of the shape of the footprint is illustrated in FIGS. 6, 7and 8. FIG. 6 shows the relative magnitude of various modes, q_(m,n)(t)for m≧n plotted against scan angle, ψ, for a square surface, when {tildeover (f)}(x, y, t) is a delta function (the plot for m≦n is the mirrorimage around ψ=45°). The angles at which each mode is a maximum areshown in the legend to the figure. FIG. 7 shows the magnitude of thecorresponding modes when {tilde over (f)}(x, y, t) is a narrow2-dimensional Caussian function of width (standard deviation) L_(x)/20.Because this is smoother than a delta function, the magnitude of themodes are lower than the corresponding modes for the delta function.FIG. 8 shows the magnitude of the modes for a wide 2-dimensionalGaussian function of width L_(x)/5 and for this case, the magnitude ofall modes is much lower, indicating that in order to avoid largedeviations in the thermal profile, the “footprint” of the gun should beas wide as possible.

Determining the Path

This analysis indicates that the thermal profile will be minimised bychoosing a scan angle such that

$\begin{matrix}{{\tan\;\psi} \neq \frac{m\; L_{y}}{n\; L_{x}}} & (91)\end{matrix}$

One such example is given in FIG. 9, which shows the pattern generatedwhen ψ=73.5° for L_(x)=L_(y), so that tan ψ=3.37. This path generates a“flat” thermal profile, but it is difficult to program-the-path into therobot as it never repeats itself, leading to a robot program that(theoretically) consists of an infinite number of points. For ease ofrobot path programming, if the robot is started at a point on one edgeof the surface, it should return to this point after a finite,manageable number of passes over the surface. Unfortunately, thecondition on the scan angle to ensure that this occurs is that

$\begin{matrix}{{\tan\;\psi} = \frac{\mu\; L_{y}}{\upsilon\; L_{x}}} & (92)\end{matrix}$where μ and ν are integers, which is exactly the same as the conditionfor exciting the thermal modes. Thus, the requirement for a closed pathis in direct contradiction to the requirement for a flat thermalprofile. This is illustrated in FIG. 10, which shows a path that repeatsitself, but it also excites the q_(5,3)(t) mode, resulting in a poorthermal profile. However, by choosing a scan angle as in FIG. 11 thatsatisfies the criterion in (92) but making sure that μ and ν aresufficiently large so that |q_(μ,ν)|_(peak) is small (because λ_(μ,ν)islarge and b_(μ,ν)is small) then a good thermal profile is achieved usinga repeating scan pattern. It is important to ensure that μ and ν have nocommon factors to avoid exciting lower order modes: for example, if μ=12and ν=6, although the magnitude of q_(12,6)(t) may be relatively small,this path will also excite q_(2,1)(t) which will be much larger.

Having chosen a scan angle, one final check that needs to be carried outis to ensure that the maximum distance between scans in the samedirection satisfies the condition for uniform mass deposition. For aspray footprint with 2-dimensional Gaussian shape, this is equivalent torequiring the that distance between the scans should be less than πσ/3,where σ is the width (standard deviation) of the Gaussian [1].

This leads to the procedure shown in FIG. 12 for determining the optimalpath.

-   -   1. Choose optimisation criterion and maximum acceptable level of        deviation from the desired thermal profile.    -   2. Input the dimensions of the surface, L_(x), L_(y) and the        scan velocity, ν.    -   3. Enter the footprint of the spray gun, {tilde over (f)}(x,        y, t) and determine the coefficients, {circumflex over        (b)}_(m,n) when the gun is centre of the surface, using (65).    -   4. Determine upper bounds, M and N, such that {{circumflex over        (b)}_(m,n)≈0: m>M;n>N}    -   5. Choose integers, μ and ν, such that μ≧ν and μ and ν have no        common factors.    -   6. Set scan angle to

$\begin{matrix}{{\tan\;\psi} = \frac{\mu\; L_{y}}{\upsilon\; L_{x}}} & (93)\end{matrix}$

-   -   7. Search over all modes, {m=1,2, . . . , M,n=1,2, . . . , N},        to ensure that all q_(m,n)(t) satisfy the optimisation criterion        for this scan angle.    -   8. If the criterion is not satisfied, increase ν and/or μ and        repeat from step 5.    -   9. If the criterion is satisfied, check that path satisfies mass        deposition criterion    -   10. If the mass deposition criterion is not satisfied, increase        ν and/or p and repeat from step 5    -   11. If the mass deposition criterion is satisfied, use scan        angle, ψ, to generate robot path and download to robot.    -   12. Stop.

If it is not possible to find a scan angle that satisfies theoptimisation criterion, then the scan velocity and/or the width of thespray footprint need to be increased until the procedure can find ansuitable path.

It should be noted that because H(t) arid {tilde over (f)}(x, y, t)depend upon z(t), their values will change as the thickness of the steelshell builds up. As a result, the optimal path may change as z(t)increases and it may be necessary to perform the optimisation at a rangeof different thicknesses.

Extensions to Other Geometries

The process described above is based upon the assumption that thesurface is flat and rectangular with edges that are insulated. Theapproach can be extended to accommodate other geometries, as follows.

-   Non-flat Surfaces The same approach can be used for surfaces that    are not flat by ensuring the height and orientation of the spray    gun(s) are adjusted so that a constant distance is maintained    between the guns and the surface and that the guns are always    oriented perpendicular to the surface. Under these circumstances,    there will be a uniform build of mass and the surface can be    considered as flat. Once the optimal scan angle, ψ, has been    determined, the robot movements required to maintain constant offset    and orientation to the surface along this path can be determined.-   Circular Surfaces The method can be adapted to accommodate surface    with circular geometry by expressing the problem in terms of    cylindrical polar co-ordinates, (r, ξ). Under these circumstances,    the spatial modes become

$\begin{matrix}{{\phi_{m,n}\left( {r,\xi} \right)} = \left\{ \begin{matrix}{{J_{0}\left( {\lambda_{m,0}r} \right)}\mspace{56mu}} & {{{{{{for}\mspace{14mu} n} = 0};{m = 1}},2,3,\ldots}\mspace{95mu}} \\{{J_{n}\left( {\lambda_{m,n}r} \right)}\cos\; n\;\xi} & {{{{for}\mspace{14mu} n} = 1},2,3,{\ldots;{m = 0}},2,4,\ldots} \\{{J_{n}\left( {\lambda_{m,n}r} \right)}\sin\; n\;\xi} & {{{{for}\mspace{14mu} n} = 1},2,3,{\ldots;{m = 1}},3,5,\ldots}\end{matrix} \right.} & (94)\end{matrix}$where J_(n)(r) are the nth order Bessel functions of the first kind andλ_(m,n) are chosen to satisfy the boundary conditions, which for thecase where the edges are insulated are

$\begin{matrix}{{\frac{\partial\phi_{m,n}}{\partial r}}_{r = {r\;}_{\max}} = 0} & (95)\end{matrix}$with r_(max) being the radius of the sprayed surface. This isparticularly relevant to controlling the thermal profile in process suchas the Osprey Process as described in P. S. Grant, “Spray Forming,”Progress in Materials Science. vol. 39. pp. 497-545, 1995, hereinincorporated by reference in its entirety, where objects with circularsymmetry are commonly formed by spray deposition. Regulating the thermalprofile during spraying in this case, controls the porosity,microstructure and yield of these processes. The approach could also beapplied to spraying onto spheres or spherical shells by expressing theproblem in terms of spherical polar co-ordinates.

-   Rotating Surfaces By transforming the co-ordinates, the thermal    profile in process where the surface and/or the spray guns are    rotated can be modelled and an optimal path found. This is    particularly applicable for processes with circular symmetry, such    as the Osprey process.-   General Shapes When spraying onto surfaces that do not have a    regular shape, it is more difficult to identify the spatial modes,    φ_(m,n)(x, y) Under these circumstances, the “long term” thermal    profile described by the partial differential equation can be    modelled using a numerical method such as finite differences as    described in K. W. Morton and D. F. Mayers, “Numerical Solution of    Partial Differential Equations,” Cambridge University Press,    Cambridge, UK 1996, or finite elements as described in K. Eriksson    et al., “Computational Differential Equations.” Cambridge University    Press. Cambridge, UK, 1996, both of these references herein    incorporated by reference in their entirety. For either method, a    suitable path can be found by defining a number of points around the    edge of the surface and then using a non-linear optimisation method,    such as simulated annealing or genetic algorithms to determine the    path between the points that minimises the thermal profile. Such    non-linear optimisation methods are described in M. H. Hassoun.    “Fundamentals of Artificial Neural Networks,” MIT Press, Cambridge,    Mass. 1995, herein incorporated by reference in its entirety.

REFERENCES

-   [1] J. K. Antonio, R. Ramabhadran, and T.-L. Ling, “A framework for    optimal trajectory planning for automated spray coating,”    International Journal of Robotics and Automation, vol. 12, no. 4,    pp. 124-134, 1997.-   [2] P. S. Grant, “Spray forming,” Progress in Materials Science,    vol. 39, pp. 497-545, 1995.-   [3] K. W. Morton and D. F. Mayers, Numerical Solution of Partial    Differential Equations, Cambridge university Press, Cambridge, UK,    1994.-   [4] K. Eriksson, D. Estep, P. Hansbo, and C. Johnson, Computational    Differential Equations, Cambridge University Press, Cambridge, UK,    1996.-   [5] M. H. Hassoun, Fundamentals of Artificial Neural Networks, MIT    Press, Cambridge, Mass., 1995.

1. A system for incrementally depositing material comprising: deliverymeans for directing material toward a deposition zone; control means,operably coupled to the delivery means, the control means forcontrolling the deposition according to a derived scan path planpredicted to reduce deviation from an ideal uniform temperature profilefor the deposition during the deposition process, wherein the derivedscan path is derived in a protocol in which at least one of thefollowing input considerations are accredited: optimization criteriaselected; maximum acceptable derivation from desired thermal profile;dimensions of deposition zone; size/dimensions or deposition footprint;scan velocity.
 2. A system according to claim 1, wherein at least one ofthe delivery means and the control means is operable to produce apattern of material deposition over the deposition zone according to thederived scan path plan.
 3. A system according to claim 1, wherein thederived scan path plan comprises substantially a mirrorbox scan pathplan.
 4. A system according to claim 1, wherein the derived scan pathplan includes a plurality of angled scan passes that intersect oneanother.
 5. A system according to claim 1, wherein the derived scan pathplan comprises reflected scan passes.
 6. A system according to claim 5,wherein the reflected scan passes have an angle of incidence to normalsubstantially equal to an angle of reflection to normal.
 7. A systemaccording to claim 5, wherein the reflected scan passes have an angle ofincidence to normal substantially different to an angle of reflection tonormal.
 8. A system according to claim 1, wherein the system has apredetermined scan angle defining the derived scan path plan.
 9. Asystem according to claim 1, wherein the derived scan path plan isrelated to the thermal footprint of the material delivered by thedelivery means.
 10. A system according to claim 9, wherein therelationship between the derived scan path plan and the thermalfootprint of the material delivered by the delivery means defines apredetermined scan angle (ψ) for the derived scan path plan.
 11. Asystem according to claim 9, wherein the relationship between thederived scan path plan and the thermal footprint of the materialdelivered by the delivery means, is such that when defining the thermalfootprint in terms of a 2-dimensional Fourier series, an optimal scanangle (ψ) is selected to avoid excitation of lower order modes.
 12. Asystem according to claim 1, wherein the material deposited is metaldelivered in-flight in molten droplet form from the delivery means. 13.A system according to claim 1, wherein the delivery means is arranged todeliver the material in-flight toward the delivery zone.
 14. A systemaccording to claim 1, wherein the delivery means comprises spraydelivery means.
 15. A system according to claim 1, wherein the deliverymeans is arranged to deliver molten droplets of material in a conveyinggas.
 16. A system according to claim 1, wherein the control meanscooperates with the deposition means to deposit material in accordancewith the a predetermined path plan having a predetermined scan rateacross the deposition zone.
 17. A system according to claim 1, whereinthe control means cooperates with deposition means to deposit materialin accordance with a predetermined path plan having a predetermined scanmovement direction.
 18. A system according to claim 1, wherein thederived scan path plan comprises a predetermined path plan derived by:i) consideration of spatial modes; and ii) selecting spatial modes tooptimise the path plan length.
 19. A system according to claim 18,wherein selection of spatial modes is conducted to avoid excitation oflower order modes.
 20. A system according to claim 1, wherein thederived scan path plan preferably reflects at boundaries to form anovercrossing pattern at the deposition zone.
 21. A system according toclaim 1, wherein the derived scan path plan comprises a repeatingpattern returning to a staff point following a plurality of scan passesover the deposition zone.
 22. A system according to claim 1, wherein thederived scan path plan comprises a non-repeating pattern.
 23. A systemaccording to claim 22, wherein a correction step operates to return thederived scan path plan to a common path point following a finite numberof scan passes.
 24. A system according to claim 1, wherein the spraydelivery means comprises a spray gun and plural axis movable positioningapparatus.
 25. A system according to claim 1, wherein the derived scanpath plan has a scan angle set in a feed back loop to determine theoptimum scan angle.
 26. A system according to claim 25, wherein anoptimum scan angle is determined in accordance with a control routine asfollows: Having regard for the footprint of the spray gun, {tilde over(f)}(x, y, t), the coefficients {circumflex over (b)}_(m,n), aredetermined; Upper bounds, M and N are determined, such that {{circumflexover (b)}_(m,n)≈0:m>M; n>N}; Integers μ, ν are selected such that μ≦Mand ν≦N and μ≧ν and μ and ν have no common factors; a scan angle (ψ) isset to ${\tan\;\psi} = \frac{\mu\; L_{y}}{{vL}_{x}}$ where L_(x) is afirst dimension in a first direction and L_(y) is a second dimension ina second direction orthogonal to the first direction; Search over allmodes, {m=1, 2, . . . , M, n=1, 2, . . . , N}to ensure that allq_(m,n)(t) satisfy the optimisation criterion for this scan angle; Ifthe criterion is not satisfied, increase ν and/or μ and repeat precedingsteps (from ‘Integers μ, ν are selected’ step); If the criterion issatisfied, check that path satisfies mass deposition criterion; If themass deposition criterion is not satisfied, increase ν and/or μ andrepeat preceding steps (from ‘Integers μ, ν are selected’ step); If massdeposition criterion is satisfied, use scan angle, ψ to generate robotpath and download to control scan.
 27. A system according to claim 1,further comprising means for monitoring the temperature history of oneor more regions of material deposited at the deposition zone.
 28. Asystem according to claim 27, wherein the control means is adapted tovary the operation of the delivery means dependent upon the monitoredtemperature history of the deposit.